The first isomorphism in Lemma REF  works for any finite unramified extension \(K\) .
The first case is immediate. For the second case, we know the restriction of \(\overline{\rho }\)  to \(\mathrm {Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)\)  is wildly ramified thanks to [1]}.
Since \(K/\mathbb {Q}_p\)  is unramified, the restriction of \(\overline{\rho }\)  to \(\mathrm {Gal}(\overline{K}/K)\)  is still wildly ramified.
Thus, it cannot be a direct sum of two characters.
Thus, we have \(E(K)[p] = 0\)  even when \(\widetilde{E}(k)[p]\)  is non-trivial.
